Brinkman number

The Brinkman number (Br) is a dimensionless number related to heat conduction from a wall to a flowing viscous fluid, commonly used in polymer processing. There are several definitions; one is

$\mathrm{Br} = \frac {\mu u^2}{\kappa(T_w-T_0)} = \mathrm{Pr} \, \mathrm{Ec}$

where

• μ is the dynamic viscosity;
• u is the flow velocity;
• κ is the thermal conductivity;
• T₀ is the bulk fluid temperature;
• T_w is the wall temperature;
• Pr is the Prandtl number
• Ec is the Eckert number

It is the ratio between heat produced by viscous dissipation and heat transported by molecular conduction. i.e., the ratio of viscous heat generation to external heating. The higher the value of it, the lesser will be the conduction of heat produced by viscous dissipation and hence larger the temperature rise.

In, for example, a screw extruder, the energy supplied to the polymer melt comes primarily from two sources:

• viscous heat generated by shear between parts of the flow moving at different velocities;
• direct heat conduction from the wall of the extruder.

The former is supplied by the motor turning the screw, the latter by heaters. The Brinkman number is a measure of the ratio of the two.

References

• J. D. Huba. NRL Plasma Formulary. Naval Research Laboratory, 1994.
• Carl W. Hall. Laws and Models: Science, Engineering and Technology. — CRC Press, Boca Raton, 2000. ISBN 978-0-8493-2018-7.
• Stephen M. Richardson. Fluid mechanics.
• L. P. Yarin, A. Mosyak, Gad Hetsroni. Fluid flow, heat transfer and boiling in micro-channels.